Probability And Statistics 11
One-Way and Two-Way ANOVA Analysis in R
This document provides comprehensive analysis of several ANOVA problems, examining differences between group means in various experimental settings. Each exercise includes detailed R code implementation and statistical interpretation.
Exercise 1: One-Way ANOVA Problems
Problem 1: Comparing Five Samples
# Problem 1: One-way ANOVA with 5 samples of unequal sizes
# Input the data
sample1 <- c(14, 13, 10, 9, 10)
sample2 <- c(10, 9, 12, 12)
sample3 <- c(11, 12, 13, 16, 17)
sample4 <- c(16, 17, 14, 13, 12, 14)
sample5 <- c(14, 12, 13)
# Combine data into a data frame
data <- data.frame(
values = c(sample1, sample2, sample3, sample4, sample5),
group = factor(c(rep("Sample1", length(sample1)),
rep("Sample2", length(sample2)),
rep("Sample3", length(sample3)),
rep("Sample4", length(sample4)),
rep("Sample5", length(sample5))))
)
# Display the data structure
str(data)
head(data)
# Perform one-way ANOVA
anova_result <- aov(values ~ group, data = data)
summary(anova_result)
# Calculate critical F-value
alpha <- 0.05
df1 <- 4 # Number of groups - 1
df2 <- 18 # Total observations - number of groups
critical_f <- qf(1 - alpha, df1, df2)
# Extract the F-value from ANOVA results
f_value <- summary(anova_result)[[^1]]["F value"][1,1]
# Decision
cat("\nObserved F-value:", f_value, "\n")
cat("Critical F-value:", critical_f, "\n")
if (f_value > critical_f) {
cat("Decision: Reject the null hypothesis.\n")
cat("Conclusion: There is a significant difference between at least one pair of group means.\n")
} else {
cat("Decision: Fail to reject the null hypothesis.\n")
cat("Conclusion: There is no significant difference between group means.\n")
}
# Visualize the data
boxplot(values ~ group, data = data,
main = "Comparison of Five Samples",
xlab = "Sample Groups", ylab = "Values",
col = c("lightblue", "lightgreen", "lightpink", "lightyellow", "lightcyan"))Problem 2: Comparing Two Samples
# Problem 2: One-way ANOVA with 2 samples
# Input the data
s1 <- c(27, 31, 31, 29, 30, 27, 28)
s2 <- c(22, 27, 25, 23, 26, 27, 23)
# Combine data into a data frame
data2 <- data.frame(
values = c(s1, s2),
group = factor(c(rep("S1", length(s1)), rep("S2", length(s2))))
)
# Display the data structure
str(data2)
head(data2)
# Perform one-way ANOVA
anova_result2 <- aov(values ~ group, data = data2)
summary(anova_result2)
# Calculate critical F-value
alpha <- 0.05
df1 <- 1 # Number of groups - 1
df2 <- 12 # Total observations - number of groups
critical_f2 <- qf(1 - alpha, df1, df2)
# Extract the F-value from ANOVA results
f_value2 <- summary(anova_result2)[[^1]]["F value"][1,1]
# Decision
cat("\nObserved F-value:", f_value2, "\n")
cat("Critical F-value:", critical_f2, "\n")
if (f_value2 > critical_f2) {
cat("Decision: Reject the null hypothesis.\n")
cat("Conclusion: There is a significant difference between the two group means.\n")
} else {
cat("Decision: Fail to reject the null hypothesis.\n")
cat("Conclusion: There is no significant difference between the two group means.\n")
}
# Visualize the data
boxplot(values ~ group, data = data2,
main = "Comparison of Two Samples",
xlab = "Sample Groups", ylab = "Values",
col = c("lightblue", "lightgreen"))Problem 3: Comparing Manager Levels
# Problem 3: One-way ANOVA for seminar evaluations by manager level
# Input the data
high_level <- c(7, 7, 8, 7, 9, 10, 8)
midlevel <- c(8, 9, 8, 10, 9, 8)
low_level <- c(5, 6, 5, 7, 4)
# Combine data into a data frame
data3 <- data.frame(
ratings = c(high_level, midlevel, low_level),
manager_level = factor(c(rep("High Level", length(high_level)),
rep("Midlevel", length(midlevel)),
rep("Low Level", length(low_level))))
)
# Display the data structure
str(data3)
head(data3)
# Perform one-way ANOVA
anova_result3 <- aov(ratings ~ manager_level, data = data3)
summary(anova_result3)
# Calculate critical F-value
alpha <- 0.05
df1 <- 2 # Number of groups - 1
df2 <- 15 # Total observations - number of groups
critical_f3 <- qf(1 - alpha, df1, df2)
# Extract the F-value from ANOVA results
f_value3 <- summary(anova_result3)[[^1]]["F value"][1,1]
# Decision
cat("\nObserved F-value:", f_value3, "\n")
cat("Critical F-value:", critical_f3, "\n")
if (f_value3 > critical_f3) {
cat("Decision: Reject the null hypothesis.\n")
cat("Conclusion: There is a significant difference in evaluations according to manager level.\n")
} else {
cat("Decision: Fail to reject the null hypothesis.\n")
cat("Conclusion: There is no significant difference in evaluations according to manager level.\n")
}
# Visualize the data
boxplot(ratings ~ manager_level, data = data3,
main = "Seminar Evaluations by Manager Level",
xlab = "Manager Level", ylab = "Evaluation Ratings (1-10)",
col = c("lightblue", "lightgreen", "lightpink"))
# Optional: Perform Tukey's HSD test to identify which groups differ
TukeyHSD(anova_result3)Exercise 2: Two-Way ANOVA Problems
Problem 1: Machines and Operators
# Problem 1: Two-way ANOVA for machines and operators
# Input the data
machine1 <- c(46, 54, 48, 46, 51)
machine2 <- c(56, 55, 56, 60, 53)
machine3 <- c(55, 51, 50, 51, 53)
machine4 <- c(47, 56, 58, 59, 55)
operator <- factor(rep(1:5, 4))
machine <- factor(rep(1:4, each = 5))
# Combine data into a data frame
data4 <- data.frame(
length = c(machine1, machine2, machine3, machine4),
operator = operator,
machine = machine
)
# Display the data structure
str(data4)
head(data4)
# Perform two-way ANOVA
anova_result4 <- aov(length ~ operator + machine + operator:machine, data = data4)
summary(anova_result4)
# Calculate critical F-values
alpha <- 0.05
# For operator
df1_operator <- 4 # Number of operators - 1
# For machine
df1_machine <- 3 # Number of machines - 1
# For interaction
df1_interaction <- 12 # (Number of operators - 1) * (Number of machines - 1)
# Residuals
df2 <- 0 # Total observations - df1_operator - df1_machine - df1_interaction
# Note: Since each operator made exactly one spacer with each machine,
# there are no degrees of freedom left for the error term (no replication)
# Therefore, we'll analyze without the interaction term
# Re-perform two-way ANOVA without interaction
anova_result4_no_interaction <- aov(length ~ operator + machine, data = data4)
summary(anova_result4_no_interaction)
# Extract F-values
f_value_operator <- summary(anova_result4_no_interaction)[[^1]]["F value"][1,1]
f_value_machine <- summary(anova_result4_no_interaction)[[^1]]["F value"][2,1]
# Calculate critical F-values
df2_no_interaction <- 12 # Residual degrees of freedom
critical_f_operator <- qf(1 - alpha, df1_operator, df2_no_interaction)
critical_f_machine <- qf(1 - alpha, df1_machine, df2_no_interaction)
# Decision
cat("\nOperator Effect:\n")
cat("Observed F-value:", f_value_operator, "\n")
cat("Critical F-value:", critical_f_operator, "\n")
if (f_value_operator > critical_f_operator) {
cat("Decision: Reject the null hypothesis for operator effect.\n")
cat("Conclusion: There is a significant difference between operators.\n")
} else {
cat("Decision: Fail to reject the null hypothesis for operator effect.\n")
cat("Conclusion: There is no significant difference between operators.\n")
}
cat("\nMachine Effect:\n")
cat("Observed F-value:", f_value_machine, "\n")
cat("Critical F-value:", critical_f_machine, "\n")
if (f_value_machine > critical_f_machine) {
cat("Decision: Reject the null hypothesis for machine effect.\n")
cat("Conclusion: There is a significant difference between machines.\n")
} else {
cat("Decision: Fail to reject the null hypothesis for machine effect.\n")
cat("Conclusion: There is no significant difference between machines.\n")
}
# Visualize the data
par(mfrow = c(1, 2))
boxplot(length ~ operator, data = data4,
main = "Spacer Length by Operator",
xlab = "Operator", ylab = "Length (mm)",
col = "lightblue")
boxplot(length ~ machine, data = data4,
main = "Spacer Length by Machine",
xlab = "Machine", ylab = "Length (mm)",
col = "lightgreen")
par(mfrow = c(1, 1))
# Interaction plot
interaction.plot(data4$machine, data4$operator, data4$length,
type = "b", col = 1:5, lty = 1, pch = 19,
main = "Interaction Plot: Machine × Operator",
xlab = "Machine", ylab = "Mean Length (mm)",
trace.label = "Operator")Problem 2: Paint Types and Steel Alloys
# Problem 2: Two-way ANOVA for paint types and steel alloys
# Input the data
paint_type <- factor(rep(1:3, 3))
steel_alloy <- factor(rep(1:3, each = 3))
resistance <- c(40, 54, 47, 51, 55, 56, 56, 50, 50)
# Combine data into a data frame
data5 <- data.frame(
resistance = resistance,
paint_type = paint_type,
steel_alloy = steel_alloy
)
# Display the data structure
str(data5)
head(data5)
# Perform two-way ANOVA
anova_result5 <- aov(resistance ~ paint_type + steel_alloy + paint_type:steel_alloy, data = data5)
summary(anova_result5)
# Calculate critical F-values
alpha <- 0.05
# For paint_type
df1_paint <- 2 # Number of paint types - 1
# For steel_alloy
df1_steel <- 2 # Number of steel alloys - 1
# For interaction
df1_interaction <- 4 # (Number of paint types - 1) * (Number of steel alloys - 1)
# Residuals
df2 <- 0 # No residual degrees of freedom since there's only one observation per cell
# Re-perform two-way ANOVA without interaction
anova_result5_no_interaction <- aov(resistance ~ paint_type + steel_alloy, data = data5)
summary(anova_result5_no_interaction)
# Extract F-values
f_value_paint <- summary(anova_result5_no_interaction)[[^1]]["F value"][1,1]
f_value_steel <- summary(anova_result5_no_interaction)[[^1]]["F value"][2,1]
# Calculate critical F-values
df2_no_interaction <- 4 # Residual degrees of freedom
critical_f_paint <- qf(1 - alpha, df1_paint, df2_no_interaction)
critical_f_steel <- qf(1 - alpha, df1_steel, df2_no_interaction)
# Decision
cat("\nPaint Type Effect:\n")
cat("Observed F-value:", f_value_paint, "\n")
cat("Critical F-value:", critical_f_paint, "\n")
if (f_value_paint > critical_f_paint) {
cat("Decision: Reject the null hypothesis for paint type effect.\n")
cat("Conclusion: There is a significant difference between paint types.\n")
} else {
cat("Decision: Fail to reject the null hypothesis for paint type effect.\n")
cat("Conclusion: There is no significant difference between paint types.\n")
}
cat("\nSteel Alloy Effect:\n")
cat("Observed F-value:", f_value_steel, "\n")
cat("Critical F-value:", critical_f_steel, "\n")
if (f_value_steel > critical_f_steel) {
cat("Decision: Reject the null hypothesis for steel alloy effect.\n")
cat("Conclusion: There is a significant difference between steel alloys.\n")
} else {
cat("Decision: Fail to reject the null hypothesis for steel alloy effect.\n")
cat("Conclusion: There is no significant difference between steel alloys.\n")
}
# Visualize the data
par(mfrow = c(1, 2))
boxplot(resistance ~ paint_type, data = data5,
main = "Corrosion Resistance by Paint Type",
xlab = "Paint Type", ylab = "Resistance",
col = "lightblue")
boxplot(resistance ~ steel_alloy, data = data5,
main = "Corrosion Resistance by Steel Alloy",
xlab = "Steel Alloy", ylab = "Resistance",
col = "lightgreen")
par(mfrow = c(1, 1))
# Interaction plot
interaction.plot(data5$steel_alloy, data5$paint_type, data5$resistance,
type = "b", col = 1:3, lty = 1, pch = 19,
main = "Interaction Plot: Steel Alloy × Paint Type",
xlab = "Steel Alloy", ylab = "Mean Resistance",
trace.label = "Paint Type")Problem 3: Tyre Types and Shock Absorber Settings
# Problem 3: Two-way ANOVA for tyre types and shock absorber settings
# Input the data
tyre_A1 <- c(5, 6, 8) # Tyre type A1 with comfort setting
tyre_A1 <- c(tyre_A1, c(8, 5, 3)) # Tyre type A1 with normal setting
tyre_A1 <- c(tyre_A1, c(6, 9, 12)) # Tyre type A1 with sport setting
tyre_A2 <- c(9, 7, 7) # Tyre type A2 with comfort setting
tyre_A2 <- c(tyre_A2, c(10, 9, 8)) # Tyre type A2 with normal setting
tyre_A2 <- c(tyre_A2, c(12, 10, 9)) # Tyre type A2 with sport setting
# Create factors for tyre type and shock absorber setting
tyre_type <- factor(rep(c("A1", "A2"), each = 9))
shock_setting <- factor(rep(rep(c("Comfort", "Normal", "Sport"), each = 3), 2))
# Combine data into a data frame
data6 <- data.frame(
roadholding = c(tyre_A1, tyre_A2),
tyre_type = tyre_type,
shock_setting = shock_setting
)
# Display the data structure
str(data6)
head(data6)
# Perform two-way ANOVA
anova_result6 <- aov(roadholding ~ tyre_type + shock_setting + tyre_type:shock_setting, data = data6)
summary(anova_result6)
# Calculate critical F-values
alpha <- 0.05
# For tyre_type
df1_tyre <- 1 # Number of tyre types - 1
# For shock_setting
df1_shock <- 2 # Number of shock settings - 1
# For interaction
df1_interaction <- 2 # (Number of tyre types - 1) * (Number of shock settings - 1)
# Residuals
df2 <- 12 # Total observations - df1_tyre - df1_shock - df1_interaction
# Extract F-values
f_value_tyre <- summary(anova_result6)[[^1]]["F value"][1,1]
f_value_shock <- summary(anova_result6)[[^1]]["F value"][2,1]
f_value_interaction <- summary(anova_result6)[[^1]]["F value"][3,1]
# Calculate critical F-values
critical_f_tyre <- qf(1 - alpha, df1_tyre, df2)
critical_f_shock <- qf(1 - alpha, df1_shock, df2)
critical_f_interaction <- qf(1 - alpha, df1_interaction, df2)
# Decision
cat("\nTyre Type Effect:\n")
cat("Observed F-value:", f_value_tyre, "\n")
cat("Critical F-value:", critical_f_tyre, "\n")
if (f_value_tyre > critical_f_tyre) {
cat("Decision: Reject the null hypothesis for tyre type effect.\n")
cat("Conclusion: There is a significant difference between tyre types.\n")
} else {
cat("Decision: Fail to reject the null hypothesis for tyre type effect.\n")
cat("Conclusion: There is no significant difference between tyre types.\n")
}
cat("\nShock Absorber Setting Effect:\n")
cat("Observed F-value:", f_value_shock, "\n")
cat("Critical F-value:", critical_f_shock, "\n")
if (f_value_shock > critical_f_shock) {
cat("Decision: Reject the null hypothesis for shock absorber setting effect.\n")
cat("Conclusion: There is a significant difference between shock absorber settings.\n")
} else {
cat("Decision: Fail to reject the null hypothesis for shock absorber setting effect.\n")
cat("Conclusion: There is no significant difference between shock absorber settings.\n")
}
cat("\nInteraction Effect:\n")
cat("Observed F-value:", f_value_interaction, "\n")
cat("Critical F-value:", critical_f_interaction, "\n")
if (f_value_interaction > critical_f_interaction) {
cat("Decision: Reject the null hypothesis for interaction effect.\n")
cat("Conclusion: There is a significant interaction between tyre type and shock absorber setting.\n")
} else {
cat("Decision: Fail to reject the null hypothesis for interaction effect.\n")
cat("Conclusion: There is no significant interaction between tyre type and shock absorber setting.\n")
}
# Visualize the data
par(mfrow = c(1, 2))
boxplot(roadholding ~ tyre_type, data = data6,
main = "Roadholding by Tyre Type",
xlab = "Tyre Type", ylab = "Roadholding",
col = "lightblue")
boxplot(roadholding ~ shock_setting, data = data6,
main = "Roadholding by Shock Absorber Setting",
xlab = "Shock Absorber Setting", ylab = "Roadholding",
col = "lightgreen")
par(mfrow = c(1, 1))
# Interaction plot
interaction.plot(data6$shock_setting, data6$tyre_type, data6$roadholding,
type = "b", col = 1:2, lty = 1, pch = 19,
main = "Interaction Plot: Shock Setting × Tyre Type",
xlab = "Shock Absorber Setting", ylab = "Mean Roadholding",
trace.label = "Tyre Type")
# Optional: Perform Tukey's HSD test to identify which groups differ
TukeyHSD(anova_result6)References
Information
- date: 2025.04.17
- time: 22:46